Let $f(x,y)=\sqrt{x}+x\sqrt{y}$. Find:

Solution (a) $f(1,4)=\sqrt{1}+1\sqrt{4}=1+2=3\qquad\hspace{5.2pt}$ $\color{#0066A7}{x=1; y=4}$.

(b)$f(a^{2},9b^{2})=\sqrt{a^{2}}+a^{2} \sqrt{9b^{2}}=a+3a^{2}b\quad \color{#0066A7}{x=a^{2}; y=9b^{2}; a>0; b>0.}$

(c) $f(x+\Delta x, y)=\sqrt{x+\Delta x}+( x+\Delta x) \sqrt{y}$

(d) $f(x,y+\Delta y)=\sqrt{x}+x\sqrt{y+\Delta y}$